Methods and Results
Simon Garnier
November 29, 2016
1 Methods
1.1 Modelling colony reproductive strategy
The relative number \(l\) of larvae in the colony at time \(t\) is modeled as a function of the form:
\[ l(t) = (1-A) \left(M-M \left(\frac{\left(\sqrt{10^{2 e}}+1\right) \sin \left(\frac{2 \pi t}{P}\right)}{\sqrt{10^{2 e} \sin ^2\left(\frac{2 \pi t}{P}\right)}+1}+1\right)\right)+M \left(\frac{\left(\sqrt{10^{2 e}}+1\right) \sin \left(\frac{2 \pi t}{P}\right)}{\sqrt{10^{2 e} \sin ^2\left(\frac{2 \pi t}{P}\right)}+1}+1\right) \]
where \(M\) represents the average of the periodical wave, \(P\) its period, and \(A\) its amplitude. The amplitude \(A\) is relative to the average \(M\) of the periodical wave. When \(A = 1\) the minimum value of the wave is 0 and the maximum is \(2 M\). When \(A = 0\) the wave is flat (i.e. its minimum and maximum values are both equal to the average of the wave). The exponent \(e\) controls the degree of “squarity” of the wave. Positive values of \(e\) return a more square-like wave while negative values return a more sine-like wave. This allows us to control how smooth the reproductive cycle is, in other words, how gradual or abrupt the transitions between brood care and reproductive phases are.
For the remainder of this study, we will arbitrarily set the value of \(P\), i.e. the length of the reproductive cycle, to 1. We will also set the value of \(M\), i.e. the average relative number of larvae in the colony, to 0.5. As a consequence, both the absolute length of the reproductive cycle and the absolute number of larvae a colony raises per reproductive cycle are constant across all comparisons. With \(P = 1\) and \(M = 0.5\) we can then simplify the previous equation as follows:
\[ l(t) = \frac{A \left(\sqrt{100^e}+1\right) \sin \left(2 \pi t\right)+\sqrt{100^e \sin ^2\left(2 \pi t\right)}+1}{2 \left(\sqrt{100^e \sin ^2\left(2 \pi t\right)}+1\right)} \]
Figure 2 shows the effect of varying either the amplitude \(A\), or the “squarity exponent” \(e\) on the temporal dynamics of the relative number of larvae present in the colony across the reproductive cycle.
Figure 2: Effect of varying (A) the amplitude \(A\) or (B) the “squarity exponent” \(e\) on the temporal dynamics of the relative number of larvae present in the colony across the reproductive cycle.
1.2 Modelling colony foraging cost
We consider three possible scenarios for the distribution of foraging costs as a function of the number of larvae that have to be fed:
“Proportional”: In this scenario, the cost of foraging grows linearly with the number of larvae. This scenario is biologically unlikely but will serve as a baseline comparison for the performance of the other two scenarios.
“High Cost of Entry”: In this scenario, the cost of foraging increases proportionally faster for smaller numbers of larvae than for larger ones. This corresponds to cases where a minimum number of workers are required before foraging yields significant benefits (for instance where ants have to overpower large prey items or other social insect colonies). This is the scenario likely faced by many ant species with army ant-like biology.
“Resource Exhaustion”: In this scenario, the cost of foraging increases proportionally slower for smaller numbers of larvae than for larger ones. This corresponds to cases where local resources are exploited faster than they are replenished, which forces workers to cover increasingly larger foraging distances as the number of larvae increases. This is the scenario that is likely faced by ant species that mainly forage as scavengers, herbivores, or individual predators, i.e. all ant species except those with army ant-like biology.
For all three scenarios, we can model the change in foraging cost \(c\) as a function of the number larvae \(l\) with a function of the form:
\[ c(l) = \frac{1}{2} (n+1) k^{1-n} l^n \]
where \(k\) is the maximum number of larvae that a colony can have at any given time, and \(n\) is a parameter that determines how the cost of foraging scales with the number of larvae to be fed. When \(n = 1\), the cost of foraging scales linearly with the number of larvae (“Proportional” scenario). When \(n > 1\), the cost of foraging grows slower for smaller than for larger numbers of larvae (“Resource Exhaustion” scenario). When \(0 \leq n < 1\), the cost of foraging grows faster for smaller than for larger numbers of larvae (“High Cost of Entry” scenario).
Note that this function is designed to ensure that its integral between 0 and \(k\) is the same regardless of the value of \(n\), hence normalizing the foraging cost between all possible values of \(n\).
For the remainder of this study, we will set \(k = 1\), which allows us to simplify the previous equation as follows:
\[ c(l) = \frac{1}{2} (n+1) l^n \]
Figure 3 shows the effect of varying \(n\) on the shape of the foraging cost function.
Figure 3: Effect of varying the scaling parameter \(n\) on the shape of the foraging cost function. The “Proportional” scenario is obtained with \(n = 1\). The “High Cost of Entry” scenario is obtained with \(0 \leq n < 1\) (\(n = 1/3\) is shown here as an example). Finally, the “Resource Exhaustion” scenario is obtained with \(n > 1\) (\(n = 3\) is shown here as an example).
1.3 Integrating reproduction strategy and foraging cost
To evaluate the performance of a given reproductive strategy under different foraging cost distributions, we calculate the total foraging cost (i.e. we integrate the composite function \(c(l(t))\)) across one entire colony cycle for different values of the relative amplitude \(A\) of the reproductive cycle, the “squarity exponent” \(e\) of the reproductive cycle, and the foraging cost scaling parameter \(n\). The general shape of the integral function is as follows:
\[ \int c(l(t)) = \frac{1}{2} (n+1) k^{1-n} \int \left((1-A) \left(M-M \left(\frac{\left(\sqrt{10^{2 e}}+1\right) \sin \left(\frac{2 \pi t}{P}\right)}{\sqrt{10^{2 e} \sin ^2\left(\frac{2 \pi t}{P}\right)}+1}+1\right)\right)+M \left(\frac{\left(\sqrt{10^{2 e}}+1\right) \sin \left(\frac{2 \pi t}{P}\right)}{\sqrt{10^{2 e} \sin ^2\left(\frac{2 \pi t}{P}\right)}+1}+1\right)\right)^n \, dt \]
With \(P = 1\), \(M = 0.5\) and \(k = 1\), we can simplify this equation as follows:
\[ \int c(l(t)) = 2^{-n-1} (n+1) \int \left(\frac{A \left(\sqrt{100^e}+1\right) \sin \left(2 \pi t\right)+\sqrt{100^e \sin ^2\left(2 \pi t\right)}+1}{\sqrt{100^e \sin ^2\left(2 \pi t\right)}+1}\right)^n \, dt\]
1.4 Software
We used Mathematica 11.0.1.0 to simplify the equations and generate the integral of the function combining the reproduction strategy with the foraging cost \(\int c(l(t))\).
We used the “integrate” function in the “stats” package (version 3.3.2) in R (version 3.3.2) to calculate the value of the integral \(\int c(l(t))\) over various values of the parameters \(A\), \(e\), and \(n\).
All figures were generated in R using the ggplot2 (version 2.2.0) and cowplot (version 0.7.0) packages.
All code used in this manuscript can be found at https://github.com/swarm-lab/projectSyncedReproduction
2 Results
2.1 Cost of foraging under different foraging scenarios and reproductive strategies
Using the integral function described above, we compute the total cost of foraging over a colony cycle for various combinations of the relative amplitude \(A\) of the reproductive cycle and the shape parameter \(n\) of the foraging cost function. Since we are not interested here in the effect of the shape of the reproductive cycle, which will be treated in the following section, we set its shape to a near-square wave (\(e = 10\)). Note, however, that the results are qualitatively equivalent with a sine wave. Figure 4 summarizes the results.
Figure 4: Total foraging cost over a reproductive cycle for various combinations of the relative amplitude \(A\) of the reproductive cycle and the shape parameter \(n\) of the foraging cost function. Values of \(n\) above the dashed line (which indicates the “Proportional” scenario) correspond to “Resource Exhaustion” scenarios, while values below the dashed line correspond to “High Cost of Entry” scenarios. Isolines (white contour lines) represent points in the parameter space with constant value, and gradient vectors (white arrows) represent the direction, but not the intensity of the local gradient.
When the foraging cost function corresponds to a “Resource Exhaustion” scenario (\(n > 1\)), the lowest total foraging cost is obtained by a non-phasic reproductive strategy that distributes the number of larvae produced by the colony uniformly in time (i.e. \(A = 0\); see top left part of Figure 4).
On the contrary, when the foraging cost function corresponds to a “High Cost of Entry” scenario (\(0 \leq n < 1\)), the total foraging cost decreases with periodical variation in the number of larvae produced by the colony. Lower costs are achieved for larger oscillation amplitudes, and colonies perform best under the most extreme phasic reproductive strategy in terms of oscillation amplitude (i.e. \(A = 1\); see bottom right part of Figure 4).
2.2 Effect of smoothness of phase transitions
Here we test the effect of abrupt phase transitions in which all larvae hatch and pupate at the exact same time (i.e. a square wave cycle) versus smooth phase transitions in which larvae hatch and pupate around an average time (i.e. a sine wave cycle) in a “High Cost of Entry” scenario. We do not test this effect in a “Resource Exhaustion” scenario because the results in the previous section show that a perfectly non-phasic reproductive strategy is favored in this case.
Using the integral function described above, we compute the total cost of foraging over a reproductive cycle for different values of the cycle’s “squarity exponent” \(e\). We set \(A\) to 1, i.e. a cycle with maximum oscillation intensity, and \(n\) to 1/4, i.e. a “High Cost of Entry” scenario. Note that results are qualitatively similar for any combination of \(A > 0\) and \(0 < n < 1\). Figure 5 summarizes the results.
Figure 5: Effect of the smoothness of phase transitions (determined by the “squarity exponent” \(e\)) on the total foraging cost over a reproductive cycle for colonies in a “High Cost of Entry” foraging scenario.
The total cost of foraging decreases with the value of the “squarity exponent” \(e\), indicating that abrupt phase transitions are more beneficial than smooth phase transitions for colonies experiencing a “High Cost of Entry” foraging scenario.